Packing Trees into the Complete Graph

Let c l 0.076122 and T1, T2,…, Tn be a sequence of trees such that mV(Ti)m l i−c(i−1). We prove that, if for each 1 l i l n there exists a vertex xi ∈ V(Ti) such that Ti−xi has at least (1−2c)(i−1) isolated vertices, then T1,…, Tn can be packed into Kn. We also prove that if T is a tree of order n+1−c′n, c′ l 1/25 (37−8 √21 ) a 0.0135748, such that there exists a vertex x ∈ V(T) and T−x has at least n(1−2c′) isolated vertices, then 2np1 copies of T may be packed into K2np1.